Crossings - Math Seminars

L'iniziativa "Crossing" mira ad accogliere in Dipartimento alcuni tra i protagonisti della matematica internazionale. Ogni ricercatore invitato terrà due seminari durante il suo soggiorno: uno focalizzato su un argomento specifico e il secondo pensato in un'ottica più ampia. Lo scopo principale è favorire collaborazioni scientifiche, promuovere l'interazione tra i giovani e la ricerca avanzata, e diffondere le idee più recenti e innovative.

In via di definizione.

•    29/02/24 - Structure and Regularity of Nonlinear Harmonic Maps
•    01/03/24 - Ricci Curvature, Fundamental Group and the Milnor Conjecture

Di seguito maggiori contenuti relativi agli incontri:

Locandina Crossings

Aaron NaberNew Horizons in Mathematics prize for 2018 and Fermat prize for 2023 F. Burgess Professor of Mathematics at Northwestern University.

Il giorno 29/02/24 presso l'Edificio U4-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), si è tenuto il 1° incontro intitolato Structure and Regularity of Nonlinear Harmonic Maps.

Abstract: We will consider harmonic maps between Riemannian manifolds u:M->N .  The first part of the talk will discuss and explain the known regularity of such mappings, in particular joint work with Daniele Valtorta on the size and rectifiability of the singular sets.  The second part of the talk will focus on sequences of such mappings u_j:M->N, where it is known that blow-up can occur on a m-2 dimensional subset.  This blow-up is characterized by the so-called defect measure, which we will review and discuss.  In recent joint work with Valtorta we have proved the energy identity, a conjectured explicit description of the defect measure in terms of bubble energy counting.


Il giorno 01/03/24 presso l’Edificio U3-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), si è tenuto il 2° incontro intitolato Ricci Curvature, Fundamental Group and the Milnor Conjecture.

Abstract: Crossings between geometry, algebra and analysis. In 1968 Milnor conjectured that there is a powerful link between Ricci curvature and the fundamental group of a manifold. After 50 years, we discuss a counterexample, because math never stops being surprising.

In particular, it was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated.  In this talk, we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.  The work is joint with Elia Brue and Daniele Semola.

There are several new points behind the result.  The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  The ability to build such a fractal structure will rely on a very twisted glueing mechanism.  Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature.  In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.